Polarization-Driven Topological Insulator Transition in a $\mathrm{GaN}/\mathrm{InN}/\mathrm{GaN}$ Quantum Well

Topological insulator (TI) states have been demonstrated in materials with a narrow gap and large spin-orbit interactions (SOI). Here we demonstrate that nanoscale engineering can also give rise to a TI state, even in conventional semiconductors with a sizable gap and small SOI. Based on advanced first-principles calculations combined with an effective low-energy $\mathbf{k}·\mathbf{p}$ Hamiltonian, we show that the intrinsic polarization of materials can be utilized to simultaneously reduce the energy gap and enhance the SOI, driving the system to a TI state. The proposed system consists of ultrathin InN layers embedded into GaN, a layer structure that is experimentally achievable.

Figure 1

(a) Counterclockwise, bands of normal semiconductors such as CdTe, bands at the zero-gap transition point, and inverted bands of semiconductors. (b) Side view of a $\mathrm{GaN}/\mathrm{InN}/\mathrm{GaN}$ QW including two atomic layers of InN in a GaN matrix along the (0001) polar orientation. Small blue balls represent N atoms, purple balls In, and blue balls Ga; the In atomic layers are marked by arrows. (c) Calculated band structure of bulk InN using the HSE hybrid functional.

Figure 2

(a) and (b) Band structure of a $\mathrm{GaN}/\mathrm{InN}/\mathrm{GaN}$ QW around the $\Gamma$ point for 2 and 4 ML of InN, based on first-principles DFT-HSE calculations. The green lines represent electron states, red lines light-hole states, and blue lines heavy-hole states. (c) Calculated energy gaps as a function of the thickness of InN layers for polar ([0001]) (blue squares) and nonpolar ($[10\overline{1}0]$) (orange diamonds) $\mathrm{GaN}/\mathrm{InN}/\mathrm{GaN}$ QWs. The thicknesses were scaled to the thickness of 1 ML of InN in a polar QW and therefore correspond to the number of InN layers in the polar case. The inset shows a $[10\overline{1}0]$ QW with two InN layers. (d) Polarization field as a function of number of inserted InN layers calculated by DFT-HSE (blue squares) and based on theoretical polarization constants (orange circles). The blue and red block arrows in the inset show the polarization direction of GaN and InN regions [19].

Figure 3

(a) Band structure of a 16 Å $\mathrm{GaN}/\mathrm{InN}/\mathrm{GaN}$ QW obtained from the six-band effective Hamiltonian. (b) Schematic of an infinite long spin Hall bar with a width (along $y$) of 1000 Å. The thickness along the [0001] growth direction (labeled as $z$) comprises the InN QW plus two 200-Å-thick GaN barrier layers on either side. The yellow lines show the helical edge states, and the green arrows show the spin orientation. The short black arrows indicate the polarization-induced electric field. (c) Band structure of the Hall bar obtained by solving the effective six-band model. The insets show the density distributions of one Kramers pair of edge states: on the left the spin-up state at $k_{\parallel }=-0.1{\AA }^{-1}$, on the right the spin-down state at $k_{\parallel }=0.1{\AA }^{-1}$. (d) Rashba spin splitting (RSS) of electron (green), HH (blue) and LH (red) subbands.